j³ zeol / j³ surf: perturbed LJ hexane
pentane
κzeol: perturbed LJ hexanepentane
butane
Figure 6.1: Parity plots: flux ratio, jzeol‡ /jsurf‡ , of all molecules affected by the parameter perturbation in AFI (left); transmission coefficients of hopping in the periodic structure,κzeol, of all affected molecules in AFI, LTL, and MFI (right).
the potential, and the results do thus not show any significant sensitivity towards small to moderate deviations in the guest-guest parameters.
6.2 Crystal Structure
Apart from the force field,147 the crystal structure represents an important input to molecular simulations of guest adsorption and diffusion in porous host materials, where it should be mentioned that all-silica zeolites are in-vestigated most frequently. Crystallographic studies on the true siliceous structure provide the atoms’ positions in the unit cell (procedure 1). If ex-perimental data are yet not available, two principle routes exist for obtaining the input structure. Either, the database of the International Zeolite Asso-ciation22 (IZA) can be accessed which, for every framework type, offers a prediction of the siliceous material on the basis of theoretical considerations (procedure 2). Or, a synthesized structure which is not purely siliceous can be converted by, for example, substituting the aluminum atoms of an alumin-osilicate by silicon atoms (procedure 3). Of course, the latter two procedures give rise to (small) structural differences from the true siliceous structure. In this context, it is interesting that a PFG NMR study by Hedin et al.148 has indicated that the self-diffusion coefficient of propene in different LTA-type
92 6 Verification of Methodology zeolites is quite sensitive to compositional and structural differences.
To investigate the influence of these structural differences on both the adsorption and the diffusion behavior of guest molecules in zeolites, Monte Carlo and reactive flux simulations of methane in different zeolite structures of the same framework type—following procedure 1, 2, and 3—have been per-formed. The simulations provided Henry coefficients, isotherms, transmission coefficients, and self-diffusion coefficients of methane in siliceous structures of the LTA, SAS, and ITE frameworks (cf., Figure 2.3). Apart from the IZA structures,22 the zeolites investigated here included the all-silica materials as experimentally determined by Corma et al. (LTA),35 Wragg et al. (SAS),65 and Cambloret al. (ITE),62as well as an aluminosilicate (LTA) by Pluth and Smith36and a magnesioaluminophosphate (SAS) by Patinecet al.66The lat-ter two are converted to purely siliceous structures according to procedure 3 mentioned above. The all-silica structures by the IZA, although published only if the framework in any composition has been confirmed by experimental means, represent hypothetical all-silica zeolites because the atomic positions are determined by refinement using fixed weighted atomic distances between nearest Si–O, O–O and Si–Si atoms. However, these structures and results obtained with them shall serve as reference in the following.
Providing an impression of how much the atomic positions in different structures of the same framework can vary, Figure 6.2 shows fractional co-ordinates of the oxygen atoms in different crystallographic projections. The plots include the per-atom deviation in fractional atomic positions,δr, of the experimental structures from the IZA structure:
δr= 1 and#ci represent the position of atom iin the unit cell while subscript “UC”
refers to the unit cell length along the respective crystallographic coordinate.
The per-atom deviation considerably exceeds 1 % only in a single case (LTA structure by Pluth and Smith36). Hence, relative atom distances seem to be usually comparable among structures of the same framework type.
Figures 6.3 and 6.4 show relative differences of the adsorption isotherms (loading θ vs pressure p) and diffusion quantities (TST self-diffusivity and transmission coefficient, respectively, vs loading) of methane in the two ex-perimentally confirmed LTA structures from the corresponding IZA values.
6.2 Crystal Structure 93
0 0.5 1
0 0.5 1
fractional coord. bˆ/bˆ UC
fractional coord. aˆ/aˆUC
fractional coord. bˆ/bˆ UC
fractional coord. aˆ/aˆUC
fractional coord. cˆ/cˆUC
fractional coord. aˆ/aˆUC
fractional coord. bˆ/bˆ UC
fractional coord. aˆ/aˆUC
fractional coord. cˆ/cˆUC
fractional coord. aˆ/aˆUC δrCamblor = 0.0124 e
IZACamblor et al.
Figure 6.2: Two-dimensional projections of fractional coordinates, ˆa/ˆaUC, ˆb/ˆbUC, and ˆc/ˆcUC, of oxygen atoms in different LTA22,35,36 (a), SAS22,65,66 (b, c), and ITE22,62 (d, e) zeolite structures.
94 6 Verification of Methodology While the adsorption differences between the structure by Cormaet al.35and the IZA structure22 are rather small, those between Pluth and Smith36 and IZA structure22 are larger; especially at low pressure (p≤0.3 bar) and low temperature (T = 200 K), where a maximal discrepancy of around −40 % is observable (Figure 6.3). Therefore, it can be concluded that the smaller the cages (Corma et al.<IZA<Pluth and Smith), the stronger is the adsorp-tion at low pressures. This is likely due to a denser and thus more attractive potential-energy field inside the smaller cages. On the other hand, the larger a cage is, the stronger is the adsorption at high pressures because the cages are simply geometrically larger and can thus accommodate more methane molecules (larger saturation loading).
Self-diffusion data were calculated at 300 K and low loadings (Figure 6.4).
In contrast to the isotherm differences, the largest discrepancies (−65 %) are observed between the structure by Cormaet al. and the IZA. Over the load-ing range studied here, this difference is practically constant, indicatload-ing that the qualitative trend ofDSTSTover loading is the same (top of Figure 6.4). Al-though the absolute value ofDTSTS (U) (diffusivity calculated from a potential-energy profile) is, for a given structure and state point, consistently larger than DSTST (diffusivity calculated from a residence histogram, i.e. from a free-energy profile) by a factor of approximately 100, the relative difference ofDSTST andDTSTS (U) between two structures is the same. This supports the assumption that entropic barriers of different structures of the same frame-work type are equal; only the potential-energy barrier changes significantly between two structures.
The obvious insensitivity of the zeolite crystal structure on the transmis-sion coefficient, as highlighted in the bottom part of Figure 6.4, stems most likely from the fact that this coefficient is mainly influenced by guest mo-lecules adsorbed in neighboring cages.116 An analysis of cage occupancy dis-tributions revealed that the disdis-tributions were equivalent for all three struc-tures at the same average loading (Figure 6.5). Hence, the transmission coefficients are the same.
To allow a comparison of the crystal structure influence between differ-ent framework types, Henry coefficidiffer-ents and TST self-diffusivities at infinite dilution were calculated because the LTA results suggested that the influ-ence was strongest at low loadings and transmission coefficients were hardly affected. All three framework types studied (LTA, SAS, ITE) belong to the group of cage-type zeolites.41 Their diffusion windows are in fact so small that a methane molecule just fits into them (kinetic diameter of a methane bead: σCH4,CH4 = 3.72 ˚A; diameter of the windows determined as the largest hard sphere that fits:149 dwind≈4 ˚A; cf., Table 6.2).
6.2 Crystal Structure 95
-50 -25 0 25
10-2 10-1 100 101 102 103
∆θ/θIZA / [%]
T = 200 K
Corma et al.
Pluth and Smith
-50 -25 0 25
10-2 10-1 100 101 102 103
∆θ/θIZA / [%]
T = 300 K
-50 -25 0 25
10-2 10-1 100 101 102 103
∆θ/θIZA / [%]
T = 600 K
pressure p / [bar]
Figure 6.3: Structure influence on adsorption (loading θ) of methane in LTA-type zeolites as a function of pressure,p, and for different temperatures, T. The values obtained from the IZA structure are taken as reference values (i.e., ∆θCorma =θCorma−θIZA).
The influence of the crystal structure on adsorption is moderate, as Table 6.2 shows. Henry coefficients differ between −32 % (LTA: Pluth and Smith) and +23 % (ITE: Camblor et al.) from the corresponding KHIZA of a given framework type. A larger Henry coefficient indicates stronger ad-sorption at low loadings and can be correlated with a smaller unit cell (cf., Appendix B) and thus cage size. In this context, the earlier conjecture that the average potential in the zeolite cage becomes denser but more attractive
96 6 Verification of Methodology
-60 -30 0 30 60
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
∆DSTST /DSTST,IZA / [%] T = 300 K
Corma et al.: DSTST(F) Corma et al.: DSTST(U)
Pluth and Smith: DSTST(F) Pluth and Smith: DSTST(U)
-10 -5 0 5 10
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
∆κ/κIZA / [%]
loading θ / [molecules cage-1]
Corma et al.
Pluth and Smith
Figure 6.4: Structure influence on the TST self-diffusivity, DTSTS , (top) and on the transmission coefficient, κ, (bottom) of methane in LTA-type zeolites as a function of loading,θ, at 300 K; analogous to Figure 6.3. Note thatDSTST was calculated on the basis of both free-energy, F, and potential-energy, U, profiles of single tagged methane molecules.
in consequence of decreasing cage size is confirmed by Figure 6.6.
Clark and Snurr134 as well as Castillo et al.150 have shown that the crys-tal structure (silicalite) can strongly influence benzene as well as water ad-sorption because of the importance of electrostatic guest-host interactions.
However, Castilloet al. have also shown that propane adsorption in silicalite was not influenced by the crystal structure.150 To sum up, it might thus be concluded that the sensitivity of adsorption to structure variations is weak as long as relatively small non-polar molecules are considered.
The impact of structure variations on the diffusivity is much stronger than on adsorption. In particular, the differences between the diffusion coef-ficients obtained with the IZA structures and the “true” all-silica crystals are
6.2 Crystal Structure 97
10-4 10-3 10-2 10-1 100
0 1 2 3 4 5 6 7
probability distribution P
cage occupancy θ / [molecules cage-1]
0.25 0.5
1 2
average loading [molecules cage-1]
IZA
Corma et al.
Pluth and Smith
Figure 6.5: Probability distribution of cage occupancy; methane in three different LTA-type zeolites at 300 K and various loadings.
large; the latter ones are smaller by factors of 2.9, 15.3, and 161 for LTA, SAS, and ITE, respectively. As has been already mentioned, the analysis of the free-energy and potential-energy profiles (Figure 6.6) revealed that the entropic diffusion barriers observed in different structures of the same frame-work type [−∆S/kB = (∆F−∆U)/(kBT)] hardly varied. Significant changes manifested only for the potential-energy barriers.
In order to better understand the structure sensitivity of the guest dif-fusivity a parameter study was conducted using the LTA zeolites. Either the unit-cell parameters (#aUC =#bUC = #cUC = lUC) were scaled proportionately or the Lennard–Jones size parameter of the guest-host interaction (σO,CH4) was changed. From the results shown in Figure 6.7 it can be concluded that enlarging the window area by increasing the unit-cell size by a given fraction (filled symbols) has precisely the same effect on the diffusion coefficient as reducing the Lennard–Jones parameter by the same fraction (open symbols which, for a given structure and thus symbol type, lie on the same trend line as the filled ones). Furthermore, the diffusivity is extremely sensitive to only small window-area changes because an enlargement of≈4 % can result in an order of magnitude larger diffusion coefficient. Most importantly, however, the differences between diffusion coefficients obtained in different structures
98 6 Verification of Methodology Table 6.2: Henry coefficients and TST self-diffusivities at zero loading of methane at 300 K. Note that the errors are given along with the diffusivities as subscripts and that the maximal relative error amounts to only 11 %.
framework type dwind KH DSTST
structure [10−10m] [10−3mol Pa−1m−3] [10−12m2s−1] LTA
IZA22 4.14 4.18 78.084.30
Cormaet al.35 4.00 4.45 27.391.60
Pluth and Smith36 4.00 2.85 112.914.05
SAS
IZA22 4.21 6.87 118.065.70
Wragget al.65 4.02 9.29 7.720.83
Patinec et al.66 4.16 6.94 160.8412.23
ITE
IZA22 4.15 56.75 155.720.89
Cambloret al.62 3.79 69.99 0.970.09
decrease with increasing window size. In other words, the structure sensit-ivity becomes the more prominent, the tighter the guest molecule fits into the window and the sensitivity will very likely be observed for any cage-type zeolite into which non-polar guest molecules diffuse.
The occurrence of window-area differences between different structures of the same topology type stem partially from different unit-cell sizes, as the (almost) parallel trends of DSTST over l2cage in the inset of Figure 6.7 suggest.
On top of this “scaling effect”, differences in individual crystal atom positions can lead to even more pronounced deviations in diffusivities. This is indicated by the discrepancies between diffusivities obtained with the 4A structure by Pluth and Smith36 and the other two LTA structures. As has been indicated earlier, the 4A structure exhibits much larger differences in atom positions from the IZA structure than the siliceous structure by Corma et al.35 does.
In comparison to the previous section, the here investigated structure ef-fect is much stronger. Because the parameter study included also variation of the guest-host Lennard–Jones interaction parameters it is clear that the
6.2 Crystal Structure 99
-5 0 5 10
0 0.2 0.4 0.6 0.8 1
normalized energy and entropy
normalized transport coordinate q / lcage F/(kBT)
U/(kBT) -S/kB
IZA
Corma et al.
Pluth and Smith
Figure 6.6: Normalized free-energy, F/(kBT), potential-energy, U/(kBT), and entropy profile, −S/kB, of a single methane molecule in LTA structures from IZA,22 Corma et al.,35 and Pluth and Smith.36 The profiles are based on residence-probability histograms across the unit cells which gave rise to slightly varying cage sizes lcageIZA, lCormacage , and lPluthcage .
results of this work are in general very sensitive to the (guest-host) force field parameters. Therefore, Section 6.5 will demonstrate that the force field real-istically describes the systems studied in this work by comparing simulation results with experimental data.
In summary, it is concluded that the self-diffusion coefficient can, in fact, be extremely sensitive to very small structural differences supporting exper-imental observations.148 This effect might have been underestimated in the past. Therefore, and because there is no standard procedure to create all-silica structures when experimental data are still lacking, it is suggested to always include the actual crystal structure data used in a simulation study, for example in the supporting information accompanying a research article or in the appendix of a monograph.
100 6 Verification of Methodology
10−12 10−11 10−10 10−9 10−8
−8 −4 0 4 8 12
self−diffusivity DSTST / [m2 s−1 ]
window−area enlargement ∆Awind/Awind,0 / [%]
Pluth & Smith: σO,CH
4 reduction Pluth & Smith: lUCPluth enlargement IZA: σO,CH
4 reduction IZA: lUCIZA enlargement Corma et al.: σO,CH
4 reduction Corma et al.: lUCCorma enlargement
10−12 10−11 10−10 10−9
140 144 148 152 lcage2 / [Å2]
Figure 6.7: Transition state theory self-diffusion coefficient, DTSTS , of methane in LTA structures from IZA,22 Corma et al.,35 and Pluth and Smith36for varying relative window-area enlargement, ∆Awind/Awind,0. Note that Awind,0 represents the window area of the original structure (i.e., with size lUC,0) and original Lennard–Jones parameter (σO,CH4,0). Thus
∆Awind/Awind,0 = [(lUC)2 − (lUC,0)2]/(lUC,0)2 for unit-cell enlargement and
∆Awind/Awind,0 =−[(σO,CH4)2−(σO,CH4,0)2]/(σO,CH4,0)2 for interaction para-meter reduction. The errors in DSTST are at maximum in the range of the largest symbol size. The inset shows the variation ofDSTST with squared cage length, l2cage; the large open circles highlight here results obtained with the original structures.